The N-icon Study
In the 1960's C.J. Roberts invented the "Sphericon". It is a peculiar object with one continuous surface and two discontinuous edges.
It is well known that there are an infinite number of N-icons. The Sphericon was built from a bicone which is two joined conic sections. The cross section of a bicone with a 90° apex is a square. If the bicone is split on this square and rejoined at a 90° turn, the Sphericon is the result. Given particular mixtures of conic sections, any cross-sectional polygon can result. For instance, a cylinder with two cones attached such that the cross section is a regular hexagon with six 120° angles, the result would be what has been termed a Hexa-sphericon.
Another way to think of N-icon generation is to take a regular polygon and sweep it 180°. If you take a square and sweep it 180° on the square's diagonal it will create a bicone. This bicone can then be cut in half yielding the square faces that allow the halves to be twisted. If the square is instead swept 180° about a line connecting the midpoints of two opposite edges, a cylinder is produced. The cylinder can be cut in half also yielding the square faces allowing twisting to occur. Sweeping on the diagonal is called the "Point Cut", and sweeping on the edge midpoints is called the "Side Cut".
My interest is in the characteristics of these higher order N-icons. To study them, I have written a program that sweeps polygons as described above to generate the Nth n-icon. In addition to generating these, I have also been able to write generalized code to color the individual surfaces of these object. This resulted in N-icons of very high order that have never been seen before.
A Sphericon has a square cross section and there for only one way to twist it. As the order of N rises the number of possible twist increments increases. For instance an order 6 N-icon has two twists. A +1 twist; and a mirror image 6-icon with a -1 twist. The 8-icon can have ±2 twist increments. The program can generate the Nth±Tth n-icon where N is the order number and T is the number of twists. If T is 0, what is generated is the base model. For instance N4+T0p would be the base model for the Sphericon; in this case a bicone.
The program generates what are called Faceted N-icons. Instead of N-icons that are made from smooth conic sections, they are generated such that the surfaces are made from multiple polygons. This is the common way to generate computer graphics models and allows these models to be imported into a number of programs. The OFF file type is used for the output file type.
Consider the base N-icon such that T is 0. One might resemble a globe with the poles at the top and bottom and it can be considered to have lines of latitude and longitude. If the N-icon were smooth it can be thought of as a faceted N-icon with an infinite number of longitudinal divisions. If these lines of longitude are ignored, a surface study will have the same results as if they were generated from smooth conic sections. Therefore surfaces will be contained by latitudinal divisions which cannot be crossed. These surfaces might also be thought of circuits or as traversal paths in graph theory.
When N-icons are twisted, the lines of latitude, or boundary lines are joined in spiral patterns. There are two types of surfaces. There are continuous surfaces such that an uninterrupted path can be traced from a starting point, back to the starting point without reversing direction or crossing a boundary. There are discontinuous surfaces such that there is no circuit back to the starting point unless direction is reversed or a boundary line is crossed.
Likewise, there are two types of edges. One being a continuous edge such that it forms a circuit and the other being discontinuous and has end points.
For every Point Cut Even Order N-icon N, there is a Even Order Side Cut version. In faceted form, when twist T is 0, a Side Cut N-icon is the dual of the Point Cut of the same N, and the two are in fact co-dual. For Odd Order N-icons, the sweeping axis will be at one vertex of the polygon and at the midpoint of the edge opposite the starting vertex. For that reason, Odd Order N-icons are both Point Cut and Side Cut at once. In the case of faceted Odd Order N-icons, when twist T is 0, they are self dual.
One problem with the faceted form of N-icons is that, when twist T is not 0, making duals does not result in the expected model. When the geometric reciprocals are taken of these, one face of the dual will span the twist plane. As the number of facets is increased this artifact is diminished. As the models of twisted N-icons become smooth, they exhibit the reciprocal properties described when the twist is 0. In addition, the reciprocal properties of Odd Order N-icons is also true of Hybrid N-icons and they too are self dual.
Even when the models are faceted the circuitry of surfaces and edge paths of Even Order Point Cut and Even Order Side Cut N-icons are co-dual. And the circuitry of surfaces and edge paths of Odd Order and Hybrid N-icons are self dual.
| The Sphericon |
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| The Sphericon: The Original N-icon. If the bicone is split on this square and rejoined at a 90° turn, the Sphericon is the result. It has one continuous surface and two discontinuous edges.
Here it is displayed as a rotating GIF and also live. (control the graphic with your mouse).
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| A Hexa-Sphericon |
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| Point Cut Even Order N-icons: When an even regular polygon is rotated around an axis formed between two opposite vertices. Pictured is a Hexa-Sphericon (one based on the hexagon). Specifically a N6+T1 N-icon. It also has one continuous surface and two discontinuous edges. Dual to the Even Order Side Cut N-icons.
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| An Octa-Sphericon Side Cut |
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| Side Cut Even Order N-icons: When an even regular polygon is rotated around an axis formed between the midpoint of two opposite edges. Pictured is an Octa-Sphericon (one based on the octagon). Specifically a N8+T1d N-icon. It also has two discontinuous surface and one continuous edge. Dual to the Even Order Point Cut N-icons.
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| A Hepta-Sphericon |
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| Odd Order N-icons: When an odd regular polygon is rotated around an axis formed between the midpoint of two opposite edges. Pictured is an Hepta-Sphericon (one based on the heptagon). Specifically a N7+T2 N-icon. It also has one discontinuous surface and one continuous edge. Self dual.
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| A 6-icon Hybrid |
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| Hybrid N-icons: When half of a Point Cut Even Order N-icon is joined with a Side Cut of the same N, a Hybrid N-icon results. Pictured is a N6+T1h Hybrid N-icon. Hybrid N-icons have at least one discontinuous surface and one discontinuous edge.
There is a case with Hybrid N-icons such that when N is a power of 2, there are no twists T such that the N-icon has more than one surface. However, all other Hybrid N-icons do have at least one twist T such that there is more than one surface. Self dual.
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| Cube with 6 Half Sphericons |
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| Polycons: When conic sections are used to augment other solids. Pictured is a cube with six half Sphericons attached.
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Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.
Credit goes to Adrian Rossiter for the generalized surface formula.
The generation of OFF and VRML files was done with
Antiprism. The
Hedron application by
Jim McNeill was used to generate VRML Switch files.
History:
2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-01 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-02-28 Open Interactive Viewer from model titles
2019-03-12 Changed email address from defunct bigfoot.com
2019-03-07 Switch from Live3D to OFF viewer
2009-03-07 Revised commentary on duals
2007-10-02 Note that Hybrids are not self dual
2007-09-06 Initial Release
Back to the main Polyhedron Page.
Link to this page as http://www.interocitors.com/polyhedra/n_icons
